The purpose of this paper, extending the results of [1]}, [2]}, is to establish the nonlinear time-asymptotic stability of planar viscous shocks for the 3-d compressible isentropic N-S equations.
For convenience, we assume a zero-mass type condition as in [3]}, [4]}, [1]} (see zero-mass-con), while the case for generic perturbations will be studied in future work. Since the perturbation in ic keeps oscillating in all directions of space, the proof is based on the construction of a suitable ansatz, \( (\tilde{\rho }, \tilde{\mathbf {m}}) = (\tilde{\rho },\tilde{\mathbf {m}})(x,t) \)  (see ansatz), which is still periodic in the transverse directions \( x^{\perp }=(x_2,x_3) \in \mathbb {T}^2 \)  and shares the same oscillations of the solution, \( (\rho ,\mathbf {m}), \)  at the far field in the normal direction \( x_1 \) . The ansatz makes it feasible to establish a framework of \( L^2(\mathbb {R}\times \mathbb {T}^2) \) -theory associated with the difference, \( (\phi ,\psi ):=(\rho ,\mathbf {m}) -(\tilde{\rho },\tilde{\mathbf {m}}) \) . More precisely, the zero-mass type condition can ensure that the perturbation associated with the density and the normal component of momentum, i.e. \( (\phi ,\psi _1) := (\rho ,\mathbf {m}_1) - (\tilde{\rho }, \tilde{\mathbf {m}}_1), \)  satisfies that
\(\int _{\mathbb {R}\times \mathbb {T}^2} (\phi ,\psi _1)(x,t) dx = 0 \qquad \forall t\ge 0.\) 
